The neighbour sum distinguishing relaxed edge colouring

A k-edge colouring (not necessarily proper) of a graph with colours in {1,2,…,k} is neighbour sum distinguishing if, for any two adjacent vertices, the sums edges incident each them are distinct. The smallest value k such that G exists denoted by χ∑e(G). When we add additional restriction must be proper, then χ∑′(G). Such colourings studied on connected at least 3 vertices. There famous conjectures these edge colourings: 1-2-3 Conjecture states χ∑e(G)≤3 G; and other χ∑′(G)≤Δ(G)+2 G≠C5. In this paper, generalize versions introducing which monochromatic set induces subgraph maximum degree most d. We call an distinguishes vertices d-relaxed colouring. denote χ∑′d(G) exists. study families graphs χ∑′ known. show number required decreases when proper condition relaxed. particular, prove χ∑′2(G)≤4 every subcubic graph. For complete graphs, χ∑′d(Kn)≤4 if d∈{⌈n−12⌉,…,n−1} also determine exact χ∑′2(Kn). Finally, χ∑′d(T) tree T.